#6749981

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,34)(28,44)(29,39)(30,41)(32,46)(33,43)(35,47)(36,40)(38,45)(42,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,27,41)(20,32,35,28)(21,36,22,29)(25,42,43,38)(26,40,37,39)(31,46,47,44)(33,45,34,48);(11,15)(13,16)(17,20,22,34,23,35,21,33)(18,25,26,31,27,43,37,47)(19,28,29,45,30,32,36,48)(24,38,40,44,41,42,39,46);(2,6)(5,7)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,33,35,34)(24,39,41,40)(25,31,43,47)(28,45,32,48)(38,46,42,44);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,29,30,36)(20,33,35,34)(24,39,41,40)(25,31,43,47)(28,45,32,48)(38,46,42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6657177);